Reduced Differential Transform Method for (2+ 1) Dimensional type of ...

Reduced Differential Transform Method for (2+1) Dimensional type of the Zakharov–Kuznetsov ZK(n,n) Equations Omer ACAN1,a) and Yıldıray KESKİN1,b) 1)

Department of Mathematics, Science Faculty, Selcuk University, Konya 42003, Turkey. a)

acan_omer@ selcuk.edu.tr and b) [email protected]

Abstract. In this paper, reduced differential transform method (RDTM) is employed to approximate the solutions of (2+1) dimensional type of the Zakharov–Kuznetsov partial differential equations. We apply these method to two examples. Thus, we have obtained numerical solution partial differential equations of Zakharov–Kuznetsov. These examples are prepared to show the efficiency and simplicity of the method.

1. INTRODUCTION Partial differential equations (PDEs) have numerous essential applications in various fields of science and engineering such as fluid mechanic, thermodynamic, heat transfer, physics [1]. Most new nonlinear PDEs do-not have a precise analytic solution. so numerical methods have largely been used to handle these equation. It is difficult to handle nonlinear part of these equations. Although most of scientists applied numerical methods to find the solution of these equations, solving such equations analytically is of fundamental importance since the existent numerical methods which approximate the solution of PDEs don’t result in such an exact and analytical solution which is obtained by analytical methods. In recent years, many researchers have paid attention to studying the solutions of nonlinear PDEs by various methods, for example, the Adomian’s decomposition method (ADM) [2-7], the variational iteration method (VIM) [8-11], the homotopy analysis method [12-14], the homotopy perturbation method (HPM) [15-17], the differential transform method (DTM) [18-21], Hirtoa’s bilinear method [22], the balance method [23], inverse scattering method [24], The RDTM was first proposed by Y. Keskin [25-28] in 2009. It has received much attention since it has applied to solve a wide variety of problems by many authors [29–33]. Zakharov–Kuznetsov (ZK) equation, (1.1) ut + auu x + (∇ 2 u ) x =0 where ∇ 2 = ∂ 2x + ∂ 2y or ∇ 2 = ∂ 2x + ∂ 2y + ∂ 2z is the isotropic Laplacian [34-36]. This means that the ZK equation is given by (1.2) ut + auu x + (u xx + u yy ) x = 0, and (1.3) ut + auu x + (u xx + u yy + u zz ) x = 0 in (2+1) and (3+1)-dimensional spaces. The ZK equation was first derived for describing weakly nonlinear ionacoustic wave in a strongly magnetized lossless plasma in two dimensions [34] A further discussion of the analytical properties of the ZK equation and some constructive results were given. Recently, in 2005, A.M. Wazwaz [37] studied the type of Zakharov–Kuznetsov equation, that is, the (2+1) dimensional and (3+1) dimensional ZK(n,n) equations of the form

ut + a (u n ) x + b(u n ) xxx + k (u n ) yyx = 0, b, k > 0

(1.4)

and (1.5) ut + a (u n ) x + b(u n ) xxx + k (u n ) yyx + r (u n ) zzx = 0, b, k , r > 0 and in 2007, C. Lin, X. Zhang, [39] studied the (3+1) dimensional modified ZK equation of the form (1.6) ut + au p u x + (u xx + u yy + u zz ) x = 0. The main purpose of this study paper has been organized as follows: Section 2 deals with the analysis of the method. In Section 3, we apply the RDTM to solve types of ZK(n,n) equations of the form (1.5) equations. Conclusions are given in Section 4.

2. ANALYSIS OF THE RDTM The basic definitions in the RDTM [25] are as follows: Definition 2.1. If function u ( x, t ) is analytic and differentiated continuously with respect to time t and space x in the domain of interest, then let  1  ∂k U k ( x) = u ( x, t ) (2.1)  k k !  ∂t  t =0 where the t-dimensional spectrum function U k ( x ) is the transformed function. In this paper, the lowercase

u ( x, t ) represent the original function while the uppercase U k ( x ) stand for the transformed function. The differential inverse transform of U k ( x ) is defined as follows: ∞

u ( x, t ) = ∑ U k ( x ) t k .

(2.2)

k =0

Then combining equation (2.1) and (2.2) we write  1  ∂k (2.3) tk .  k u ( x, t )  k = 0 k !  ∂t t = 0 From the above definitions, it can be found that the concept of the reduced differential transform is derived from the power series expansion. For the purpose of illustration of the methodology to the proposed method, we write the ZK(n,n) equation in the standard operator form (2.4) L ( u ( x, t ) ) + N ( u ( x, t ) ) = 0 with initial condition (2.5) u ( x, y, 0 )= f ( x, y ) ∞

u ( x, t ) = ∑

∂ is a linear operator which has partial derivatives, N ( u ( x, t ) ) =a (u n ) x + b(u n ) xxx + k (u n ) yyx is a ∂t nonlinear term. TABLE 1. The fundamental operators of RDTM [25-28] Functional Form Transformed Form

where L =

w= ( x, y , t ) u ( x, y , t ) ± α v ( x, y , t )

 1  ∂k  k u ( x, y , t )  k !  ∂t t =0 W= U k ( x, y ) ± αVk ( x, y ) ( α is a constant) k ( x, y )

w ( x, y , t ) = x m y n t p u ( x, y , t )

Wk ( x, y ) = x m y nU ( k − p ) ( x, y )

U k ( x, y ) =

u ( x, y , t )

Wk ( x, y ) w ( x, y, t ) = u ( x, y, t ) v ( x, y, t= )

k

k

= V ( x, y )U ( x, y ) ∑ U ∑

r k −r r 0= r 0 =

r

( x, y )Vk − r ( x, y )

(k + r )! ∂r (k 1)...(k + r )U k +1 ( x, y ) = Wk ( x, y ) =+ U k + r ( x, y ) u ( x, y , t ) r k! ∂t ∂ ∂ Wk ( x, y ) = m n U k ( x, y ) w( x, y, t ) = m n u ( x, y, t ) ∂x ∂y ∂x ∂y According to the RDTM and Table 1, we can construct the following iteration formula: (k + 1)U k +1 ( x) = − N (U k ( x) )

w( x, y, t ) =

(2.6)

where N k = N (U k ( x) ) is the transformations of the function N ( u ( x, t ) ) respectively. For the easy to follow of the reader, we can give the first few nonlinear term are ∂ ∂3 ∂3 N 0 =a U 0n ( x, y ) + b 3 U 0n ( x, y ) + k 2 U 0n ( x, y ) ∂x ∂x ∂y ∂x N1 =a = N2 a

∂ ∂3 ∂3 n U0( n −1) ( x, y )U1 ( x, y ) ) + b 3 ( n U0( n −1) ( x, y )U1 ( x, y ) ) + k 2 ( n U0( n −1) ( x, y )U1 ( x, y ) ) ( ∂x ∂x ∂y ∂x ∂ n  ( n − 2) 2 ( n −1) ( n − 2) 2  ( n U0 ( x, y )U1 ( x, y ) + n U0 ( x, y )U 2 ( x, y ) − U 0 ( x, y )U1 ( x, y ) )  ∂x  2 

+b

∂3  n  n U0( n − 2) ( x, y )U12 ( x, y ) + n U0( n −1) ( x, y )U 2 ( x, y ) − U 0( n − 2) ( x, y )U12 ( x, y ) )  ( 3  ∂x  2 

∂3  n  ( n − 2) 2 ( n −1) ( n − 2) 2  ( n U0 ( x, y )U1 ( x, y ) + n U0 ( x, y )U 2 ( x, y ) − U 0 ( x, y )U1 ( x, y ) )  ∂y 2 ∂x  2  Maple Code for Nonlinear Function as given [25] restart; NF:=Nu(t,x,y)^p:#Nonlinear Function m:=4: # Order u[t]:=sum(u[b]*t^b,b=0..m): NF[t]:=subs(Nu(x,t)=u[t],NF): s:=expand(NF[t],t): dt:=unapply(s,t): for i from 0 to m do n[i]:=((D@@i)(dt)(0)/i!): print(N[i],n[i]); #Transform Function od: From initial condition (1.2), we write (2.7) U 0 ( x, y ) = f ( x, y ) Substituting (2.7) into (2.6) and by a straight forward iterative calculations, we get the following U k ( x, y ) values. +k

Then the inverse transformation of the set of values {U k ( x, y )}k = 0 gives approximation solution as, n

n

un ( x, y, t ) = ∑ U k ( x, y )t k

(2.8)

k =0

where n is order of approximation solution. Therefore, the exact solution of problem is given by u ( x, y, t ) = lim un ( x, y, t ) . n →∞

3.

(2.9)

NUMERICAL APPLICATIONS

In this section, we test the RDTM for the ZK(3,3) and ZK(2,2) equations whit fully nonlinear dispersion. Numerical results are very encouraging. Example 3.1. First we consider the following ZK(3,3) equations [38]: ut + (u 3 ) x + 2 u 3() xxx + 2 u 3() yyx = 0 (3.1)

subject to initial condition 3 1 λ sinh ( x + y ) 2 6 Taking differential transform of (3.1) and the initial condition (3.2) respectively, we obtain  1  ∂ 3 ∂3 3 ∂3 U k +1 ( x, y ) = U x U x U 03 ( x)  ( ) 2 ( ) 2 − + +  0 0 3 2 (k + 1)  ∂x ∂x ∂y ∂x  = u ( x, y, 0)

where the t-dimensional spectrum function U k ( x, y ) are the transformed function. From the initial condition (3.2) we write 3 1 U 0 ( x, y ) = λ sinh ( x + y ). 2 6 Now, substituting (3.4) into (3.3), we obtain the following U k ( x, y ) values successively

(3.2)

(3.3)

(3.4)

 3  x + y  3  x+ y U 1 ( x, y ) = − λ 3 cosh  − 8  9 cosh    8  6   6    3 2  x + y   x+ y 4  x+ y U 2 ( x, y ) = − 729 cosh 2  + 91 λ sinh   765 cosh     64  6   6   6     x + y  4  x+ y + 188181cosh 6   −382293 cosh    1 7  x + y   6   6  U 3 ( x, y ) = − λ cosh    256  6  2  x+ y − 39851  +234468 cosh     6      8  x+ y 6  x+ y  93534345 cosh    − 198626022 cosh  6  6 x y + 1        U 4 ( x, y ) = λ 9 sinh    4096 x y x y + +  6      4 2  +135212355 cosh   − 30715929 cosh  6  + 1179946   6      (3.5)

From (2.8) 4

u4 ( x, y, t ) = ∑ U k ( x, y )t k

(3.6)

k =0

Substituting (3.4) and (3.5) in (3.6), we have    x + y  − 13824 cosh 3  x + y  + 12288λ 2 t cosh  x + y  2       6144λ t sinh    6   6   6    x+ y x+ y     4 2 4 x+ y 4 2 2 x+ y  +146880λ t sinh  6  cosh  6  − 139968λ t sinh  6  cosh  6      +17472λ 4 t 2 sinh  x + y  + 6116688λ 6 t 3 cosh 5  x + y  − 3010896λ 6 t 3 cosh 7  x + y           1 6  6  6     λ u4 ( x, y , t ) =  + + + + x y x y x y x y 4096   + 637616λ 9 t 3 cosh   + 93534345λ 8 t 4 sinh   cosh8   6 3 3 −3751488λ t cosh           6   6   6   6    x+ y  x + y  cosh 4  x + y  6 x+ y 8 4  −198626022λ 8 t 4 sinh    cosh   + 135212355λ t sinh        6   6   6   6    x+ y x+ y   8 4 2 x+ y 8 4  −30715929λ t sinh    cosh   + 1179946λ t sinh    6   6   6   

when analyzed the solutions of ZK(3,3) equations by RDTM, the following results are obtained: TABLE 1. For ZK(3,3), comparison of absolute error of the numerical results for u4 ( x, y, t ) , by RDTM λ

0.00001

x

y

0.0 0.0

0.0 0.5

−0.375000000 × 10

0.0

1.0

0.2511590160 × 10 −

0.5 0.5

0.0 0.5

0.5 1.0

t

RDTM Solution

0.2265468750 × 10

−37

−5

0.2292047232 × 10

−37

0.1251447262 × 10

0.001

1.0 0.0

Abs-Error-RDTM

−18

5

0.2373142696 × 10

−37

0.1251447262 × 10

−5

0.2292047232 × 10

−37

0.2511590160 × 10

−5

0.2373142096 × 10 −37

0.3789184752 × 10

−5

0.2512918476 × 10 −37

0.2511590160 × 10

−5

0.2373142696 × 10 −37 0.2512918476 × 10 −37

0.2718596318 × 10

1.0

0.5

0.3789184752 × 10

−5

1.0

1.0

0.5093108360 × 10

−5

Example 3.2. Now we consider the following ZK(2,2) equation: 1 1 0 ut + (u 2 ) x + (u 2 ) xxx + (u 2 ) yyx = 8 8 subject to initial condition 4 u ( x, y, 0) = − λ cosh 2 ( x + y ) 3 Similarly by using the RDTM and Table 1 to (3.7) and (3.8) , we obtain the recursive relation  1  ∂ 2 1 ∂3 2 1 ∂3 − U k +1 ( x, y ) = U 0 ( x) + U 02 ( x)   U 0 ( x) + 3 (k + 1)  ∂x 8 ∂x 8 ∂y 2 ∂x  where the t-dimensional spectrum function U k ( x, y ) are the transformed function. From the initial condition (3.8) we write 4 U 0 ( x, y ) = − λ cosh 2 ( x + y ) 3 Now, substituting (3.10) into (3.9), we obtain the following U k ( x, y ) values successively

−37

(3.7)

(3.8)

(3.9)

(3.10)

32 U 1 ( x, y ) = − λ 2 cosh ( x + y ) sinh ( x + y ) (10 cosh 2 ( x + y ) − 3) 9 64 U 2 ( x, y ) = − λ 3 (1200 cosh 6 ( x + y ) − 1520 cosh 4 ( x + y ) + 408 cosh 2 ( x + y ) − 9 ) 27  23800 cosh 6 ( x + y ) − 28500 cosh 4 ( x + y )  4096 4 U 3 ( x, y ) = λ cosh ( x + y ) sinh ( x + y )  −   +8265 cosh 2 ( x + y ) − 423  243   2 8 10 1024 5  −11151 + 1512792 cosh ( x + y ) − 124257600 cosh ( x + y ) + 58864000 cosh ( x + y )  U 4 ( x, y ) = − λ    −21520320 cosh 4 ( x + y ) + 85809600 cosh 6 ( x + y )  729   (3.11) From (2.8) 4

u4 ( x, y, t ) = ∑ U k ( x, y )t k k =0

Substituting (3.10) and (3.11) in (3.12), we have

(3.12)

 243 cosh 2 ( x + y ) + 6480λ t cosh 3 ( x + y ) sinh ( x + y ) − 656640λ 2 t 2 cosh 4 ( x + y )    2 2 2 2 2 3 3 7  +176256λ t cosh ( x + y ) + 3888λ t + 73113600λ t cosh ( x + y ) sinh ( x + y )    3 3 5 3 3 3 4  −87552000λ t cosh ( x + y ) sinh ( x + y ) + 25390080λ t cosh ( x + y ) sinh ( x + y )  λ u4 ( x, y, t ) = 729  −1299456λ 3 t 3 cosh ( x + y ) sinh ( x + y ) − 2854656λ 4 t 4 + 387274752λ 4 t 4 c osh 2 ( x + y )     −31809945600λ 4 t 4 c osh 8 ( x + y ) + 15069184000λ 4 t 4 c osh10 ( x + y )     −5509201920λ 4 t 4 c osh 4 ( x + y ) + 21967257600λ 4 t 4 c osh 6 ( x + y )   

when analyzed the solutions of ZK(2,2) equations by RDTM, the following results are obtained: TABLE 2. For ZK(2,2), comparison of absolute error of the numerical results for u4 ( x, y, t ) , by RDTM λ

0.00001

x

y

0.0

0.0

-0.00001333333333

0.2932786015 × 10 −

0.0 0.0

0.5 1.0

-0.00001695387292 -0.00003174798469

0.2692669148 × 10 −20

t

RDTM Solution

Abs-Error-RDTM 24

0.1562939098 × 10 −

19

−20

-0.00001695387292

0.2692669148 × 10

-0.00003174798469 -0.00007378450649

0.1562939098 × 10 −

0.0

-0.00003174798469

0.1562939098 × 10 −19

0.5 1.0

-0.00007378450649 -0.00001887222243

0.1005498720 × 10 −

0.5

0.0

0.5 0.5

0.5 1.0

1.0 1.0 1.0

0.001

19

0.1005498720 × 10

−18

18

0.6999706576 × 10

−18

Now for a better understanding, two examples given above are as follows absolute error graphics:

FIG. 1: Absolute errors for (a) ZK(3,3) and (b) ZK(2,2) equations when -2 ≤ x ≤ 2, - 2 ≤ y ≤ 2, t = 0,001 and λ = 0.00001.

4. CONCLUSION In this study, we apply RDTM on (2+1) dimensional ZK(3,3) and ZK(2,2) PDEs with fully nonlinear dispersion. The obtained result which obtained are highly reliable an encouraging.

REFERENCES 1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26.

L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, 1997. G. Adomian, A new approach to nonlinear partial differential equations, J. Math. Anal. Applic. 102 (1984) 420–434. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, 1994. D. Kaya, M. Aassila, An application for a generalized KdV equation by the decomposition method, Physics Letters A 299 (2002) 201–206. D.D. Ganji, E.M.M. Sadeghi, M.G. Rahmat, Modi fied Camassa –Holm and Degasperis–Procesi Equations Solved by Adomian’s Decomposition Method and Comparison with HPM and Exact Solutions, Acta Appl Math 104 (2008) 303–311. A.M. Wazwaz, Partial differential equations: methods and applications. The Netherlands: Balkema Publishers, 2002. J.H. He,Anewapproach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 203–205. J.H. He, Variational iteration method-a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics 34 (4) (1999) 699–708. A.T. Abassy, M. A. El-Tawil, H. El. Zoheiry, Solving nonlinear partial differential equations using the modified variational iteration Padé technique, Journal of Computational and Applied Mathematics 207 (1) (2007) 73-91. J. Biazar, H. Ghazvini, He’s variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear Sci. Numer. Simul. 8 (3) (2007) 311–314. D.D. Ganji, A. Sadighi, I. Khatami, Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences, Physics Letters A 372 (2008) 4399–4406. S.J. Liao, Homotopy analysis method: a new analytic method for nonlinear problems, Applied Mathematics and Mechanics (English-Ed.) 19 (10) (1998) 957–962. S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003. S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2) (2004) 499– 513. J.H. He, Homotopy perturbation method: A new nonlinear technique, Appl. Math. Comput. 135 (2003) 73–79. J. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151, 287–292 (2004). J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Internat J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 207–208. J.K. Zhou, Differential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986. N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation 7 (1) (2006) 65–70. A. Kurnaz, G. Oturanc, M.E. Kiris, n-Dimensional differential transformation method for solving linear and nonlinear PDE’s, International Journal of Computer Mathematics 82 (2005) 369–80. F. Ayaz, On the two-dimensional differential transform method, Applied Mathematics and Computation 143 (2003), 361–74. X.-B. Hu, Y.-T. Wu, Application of the Hirota bilinear formalism to a new integrable differential- difference equation, Physics Letters A 246 (6) (1998) 523–529. M. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A 216 (1–5) (1996) 67–75. V.O. Vakhnenko, E.J. Parkes, A.J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalized Vakhnenko equation, Chaos Solitons Fractals 17 (4) (2003) 683–692. Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 10 (6) (2009) 741–749. Y. Keskin, G. Oturanc, The reduced differential transform method: a new approach to factional partial differential equations, Nonlinear Sci. Lett. A 1 (2) (2010) 207–217.

27. Y. Keskin, G. Oturanc, Reduced differential transform method for generalized KdV equations, Math. Comput. Applic. 15 (3) (2010) 382–393. 28. Y. Keskin, Ph.D Thesis, Selcuk University, 2010 (in Turkish). 29. A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation, J. Egypt.Math. Soc. 21 (3) (2013) 259–265. 30. R. Abazari, M. Abazari, Numerical study of Burgers–Huxley equations via reduced differential transform method, Comput. Appl. Math. 32 (1) (2013) 1–17. 31. B. bis, M. Bayram, Approximate solutions for some nonlinear evolutions equations by using the reduced differential transform method, Int. J. Appl. Math. Res. 1 (3) (2012) 288–302. 32. R. Abazari, B. Soltanalizadeh, Reduced differential transform method and its application on Kawahara equations, Thai J. Math. 11 (1) (2013) 199–216. 33. M.A. Abdou, A.A. Soliman, Numerical simulations of nonlinear evolution equations in mathematical physics, Int. J. Nonlinear Sci. 12 (2) (2011) 131–139. 34. V.E. Zakharov, E.A. Kuznetsov, On three-dimensional solitons, Sov. Phys. 39 (1974) 285–288. 35. S. Monro, E.J. Parkes, The derivation of a modified Zakharov–Kuznetsov equation and the stability of its solutions, J. Plasma Phys. 62 (3) (1999) 305–317. 36. S. Monro, E.J. Parkes, Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation, J. Plasma Phys. 64 (3) (2000) 411–426. 37. A.M. Wazwaz, Nonlinear dispersive special type of the Zakharov–Kuznetsov equation ZK(n,n) with compact and non-compact structures, Appl. Math. Comput. 161 (2005) 577–590. 38. T. Nawaz, A. Yıldırım, S. T. Mohyud-Din, Analytical solutions Zakharov–Kuznetsov equations, Advanced Powder Technology, 24 (2013) 252–256. 39. C. Lin, X. Zhang, The formally variable separation approach for the modified Zakharov–Kuznetsov equation, Communications in Nonlinear Science and Numerical Simulation.12 (2007) 636-642